The graph of a sinusoidal function has a maximum point at $(0,5)$ and then has a minimum point at $(2\pi,-5)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Answer: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline passes exactly between the maximum value $5$ and the minimum value $-5$, so the midline equation is $y={0}$. The extremum points are $5$ units above or below the midline, so the amplitude is ${5}$. The minimum point is $2\pi$ units to the right of the maximum point, so the period is $2\cdot 2\pi={4\pi}$. [Why did we multiply by 2?] Determining the type of function to use Since the graph has an extremum point at $x=0$, we should use the cosine function and not the sine function. This means there's no horizontal shift, so the function is of the form $a\cos(bx)+d$. [How do we know that?] Determining the parameters in $a\cos(bx)+d$ Since the extremum point at $x=0$ is a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${5}$, so $|a|={5}$. Since $a>0$, we can conclude that $a=5$. The midline is $y={0}$, so $d=0$. The period is ${4\pi}$, so $b=\dfrac{2\pi}{{4\pi}}=\dfrac{1}{2}$. The answer $f(x)=5\cos\left(\dfrac{1}{2}x\right)$